Offset mirrored geometry
Actually a linked duplication of your object would be simpler than any solution: Duplicate your object with AtlD, flip it on the Y axis with SY-1 and move it so that the 2 objects cross each others.
But other than that I was able to do it with the Array modifier. It looks like you can't shift the object with the Mirror modifier(?).

Create an empty close to your object.



Give your object an Array modifier, disable Relative Offset, enable Object Offset with the empty as Object.



Select your empty, flip it on the Y axis with SY-1. The object is mirrored on the Y axis.



Move the empty so that the objects cross the way you want.